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Global heat kernel estimates for symmetric Markov processes dominated by stable-like processes in exterior $C^{1,η}$ open sets

In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,η}$ open sets for all $t> 0$. The processes are symmetric pure jump Markov processes with jumping kernel intensity $$κ(x, y)ψ(|x-y|)^{-1}|x-y|^{-d-α}$$ where $α\in(0,2)$, $ψ$ is an increasing function on $[ 0, \infty)$ with $ψ(r)=1$ on $0<r\le 1$ and $c_1e^{c_2r^β}\le ψ(r)\le c_3e^{c_4r^β}$ on $r>1$ for $β\in[0, \infty]$. A symmetric function $κ(x, y)$ is bounded by two positive constants and $|κ(x, y)-κ(x,x)|\le c_5 |x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. As a corollary of our main result, we estimates sharp two-sided Green function for this process in $C^{1,η}$ exterior open sets.